Problem: Simplify the following expression and state the condition under which the simplification is valid. $q = \dfrac{9p^2 - 9p - 504}{4p^3 - 28p^2 - 32p}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ q = \dfrac {9(p^2 - p - 56)} {4p(p^2 - 7p - 8)} $ $ q = \dfrac{9}{4p} \cdot \dfrac{p^2 - p - 56}{p^2 - 7p - 8} $ Next factor the numerator and denominator. $ q = \dfrac{9}{4p} \cdot \dfrac{(p - 8)(p + 7)}{(p - 8)(p + 1)}$ Assuming $p \neq 8$ , we can cancel the $p - 8$ $ q = \dfrac{9}{4p} \cdot \dfrac{p + 7}{p + 1}$ Therefore: $ q = \dfrac{ 9(p + 7)}{ 4p(p + 1)}$, $p \neq 8$